Applications of Complex Numbers

• Applications of complex numbers

• Ideas for Math IA

• I’ve been interested in this for a while.

• There are various topics in Physics, but they seem too complicated to handle for Math IA, so I’ll skip them.

  • I’ll use them as references.
  • They are mostly used to simplify rotation calculations (takker).
  • The only situations where complex numbers are essential should be in quantum mechanics.

• I want to find more relatable topics.

• https://enjoymath.pomb.org/?p=1093

  • Model of rotational motion (converting angles to coordinates)

• Can it be connected to circular time?

  • Like seasons or a week
  • Similar to n-color Function

• It would be great if I could use generalization of Euler’s Form for complex numbers

  • It can be expressed as 1^(percentage of angle), which is nice.
  • The angle can be represented as a value between 0 and 1.
    • Probability, perhaps?
  • Adding angles represented by values between 0 and 1
  • Not just powers of 1, but powers of n
    • For example, if we model something in a year, we can directly represent the date as the exponent.
  • Well, I can just convert it to radians, but it’s an interesting idea.

• I want to consider not only two-dimensional complex numbers, but also three-dimensional ones.

  • It’s impossible to extend to three dimensions, but we can extend it to $2^n$ dimensions (takker).
    • Quaternion is famous for that.
  • It might be interesting to think about the relationship between spherical coordinates and quaternion.
  • I see (blu3mo).

• https://azisava.sakura.ne.jp/mandelbrot/definition.html

  • Mandelbrot Set